Bab 8 Pemodelan Matematik
Matematik Tingkatan 5 · 22 lessons
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" ## Sukatan Serakan Data Terkumpul ### Definisi Ukuran serakan menerangkan taburan atau kepelbagaian data dalam satu set data. Untuk data berkelompok, ukuran ini menunjukkan sejauh mana titik data menyimpang daripada nilai pusat seperti min. - **Serakan**: Merujuk kepada sejauh mana nilai data tersebar atau berkumpul. - **Data Berkelompok**: Data yang diatur ke dalam kelas atau selang, diwakili oleh jadual taburan kekerapan. ### Konsep Utama - **Julat (Range)**: Perbezaan antara nilai tertinggi dan terendah dalam set data. - **Varians**: Purata sisihan kuasa dua daripada min, mewakili taburan titik data. - **Sisihan Piawai**: Akar kuasa dua varians, menunjukkan purata jarak titik data daripada min. - **Sisihan Min**: Purata sisihan mutlak daripada min atau median. ### Info Penting - **Julat**: $$Julat = Titik Tengah Kelas Tertinggi - Titik Tengah Kelas Terendah$$ - **Varians untuk Data Berkelompok**: $$^2 = fx^2{ f} - x^2$$ di mana $f$ ialah kekerapan kelas, $x$ ialah titik tengah kelas, dan $x$ ialah min. - **Sisihan Piawai**: $$ = { fx^2{ f} - x^2}$$ ### Formula Asas - **Min untuk Data Berkelompok**: $$x = fx{ f}$$ - **Varians untuk Data Berkelompok**: $$^2 = fx^2{ f} - x^2$$ - **Sisihan Piawai**: $$^2 = fx^2{ f} - x^2$$ ### Contoh - **Contoh Asas**: Hitung min untuk data berkelompok dengan titik tengah $x = {10, 20, 30}$ dan kekerapan $f = {4, 5, 6}$. $$x = 4 10 + 5 20 + 6 30{4 + 5 + 6} = 40 + 100 + 180{15} = 320{15} 21.33$$ - **Aplikasi Lanjutan**: Cari sisihan piawai untuk data yang sama: $$^2 = fx^2{ f} - x^2$$ ### Kesilapan Lazim - Salah mengira titik tengah $x_i$ untuk data berkelompok. - Lupa menggunakan $f_i$ semasa mengira min atau varians untuk data berkelompok. - Keliru antara varians dan sisihan piawai. ### Topik Berkaitan - **Ukuran Kecenderungan Pusat** - **Taburan Kekerapan** ### Soalan Ulangkaji Pantas - Bagaimana anda mengira varians untuk data berkelompok? - Apakah perbezaan antara varians dan sisihan piawai? "
Sukatan Serakan Data Terkumpul
Selesaikan untuk $ x $ dalam persamaan $ 4x + 7 = 19 $.
Correct Answer: A (3)
- Subtract 7 from both sides: 4x + 7 - 7 = 19 - 7 → 4x = 12
- Divide both sides by 4: x = 12 ÷ 4 = 3
- Check: 4(3) + 7 = 12 + 7 = 19 ✓
Why the others are wrong:
- B (2): 4(2) + 7 = 15, not 19
- C (4): 4(4) + 7 = 23, not 19
- D (5): 4(5) + 7 = 27, not 19
Tafsirkan graf di bawah untuk mencari pintasan-x garis tersebut.
# Explanation
Why A (-3) is correct:
The x-intercept is where a line crosses the x-axis (where y = 0). Looking at the graph, the line crosses the x-axis at the point x = -3.
Why the others are wrong:
- B (3): This would be on the positive side of the x-axis, but the line crosses on the negative side.
- C (0): This is the origin, not where this line crosses the x-axis.
- D (-1): While this is on the negative x-axis, the line doesn't cross there—it crosses further left at -3.
Jika $ y = 2x^2 - 4x + 1 $, cari $ y $ apabila $ x = 3 $.
Correct Answer: A. 7
Substitute x = 3 into y = 2x² - 4x + 1:
- y = 2(3)² - 4(3) + 1
- y = 2(9) - 12 + 1
- y = 18 - 12 + 1 = 7 ✓
Why others are wrong:
- B (9): You'd get this if you forgot to multiply 2 × 9 and instead calculated 9 - 12 + 1
- C (15): This happens if you skip the -4x term entirely
- D (13): You'd get this if you calculated 18 - 4 - 1 (mixing up the order of operations)
Selesaikan untuk $ x $ dalam persamaan $ 2x^2 - 8 = 0 $.
A. 2 dan -2 ✓
Rearrange: 2x² = 8 → x² = 4 → x = ±2. When you substitute back, 2(2)² - 8 = 8 - 8 = 0 ✓
B. 4 dan -4 — Too large; 2(4)² - 8 = 32 - 8 = 24 ≠ 0
C. 1 dan -1 — Too small; 2(1)² - 8 = 2 - 8 = -6 ≠ 0
D. 3 dan -3 — Wrong value; 2(3)² - 8 = 18 - 8 = 10 ≠ 0
Persamaan $ y = x^2 $ mewakili jenis fungsi apa?
Correct Answer: Kuadratik
Persamaan y = x² adalah fungsi kuadratik karena variabel x memiliki pangkat tertinggi 2. Fungsi kuadratik selalu berbentuk y = ax² + bx + c dan menghasilkan grafik parabola.
Mengapa opsi lain salah:
- Linear: Fungsi linear memiliki pangkat 1 (seperti y = 2x + 3), bukan pangkat 2
- Eksponen: Fungsi eksponen memiliki variabel di pangkat (seperti y = 2^x), bukan pangkat sebagai konstanta
- Logaritma: Fungsi logaritma adalah kebalikan dari eksponen (seperti y = log x), tidak cocok untuk y = x²
Graf $ y = 2x + 3 $ melalui titik manakah antara pilihan berikut?
Correct answer: A (1, 5)
Substitute x = 1 into the equation: y = 2(1) + 3 = 5. So the point (1, 5) lies on the line. ✓
Why others are wrong:
- B (2, 6): When x = 2, y = 2(2) + 3 = 7, not 6
- C (0, 2): When x = 0, y = 2(0) + 3 = 3, not 2 (the y-intercept is 3)
- D (1, 4): When x = 1, y = 5, not 4
Lukis graf $ y = -x^2 + 4 $. Kenal pasti titik maksimumnya.
Why A is correct:
The equation y = -x² + 4 is a downward-opening parabola (negative coefficient on x²). The maximum point occurs at the vertex. Since there's no linear term (bx), the x-coordinate of the vertex is 0. When x = 0: y = -(0)² + 4 = 4, so the maximum point is (0, 4).
Why others are wrong:
- B & D: These points have y = 0, which are the x-intercepts (where the parabola crosses the x-axis), not the maximum.
- C: This point (4, 0) doesn't even lie on the parabola—it's not relevant to this graph.
Jika $ y $ berubah langsung dengan $ x $ dan $ y = 10 $ apabila $ x = 2 $, cari $ y $ apabila $ x = 5 $.
# Penjelasan
Jawaban yang benar: A. 25
• Perubahan langsung berarti y = kx (k adalah konstanta)
• Dari y = 10 ketika x = 2, kita cari k: 10 = k(2), jadi k = 5
• Dengan k = 5, ketika x = 5: y = 5 × 5 = 25 ✓
Mengapa pilihan lain salah:
- B. 20: Hasil dari 10 × 2 (mengalikan nilai awal, bukan menggunakan rumus yang benar)
- C. 15: Tidak sesuai dengan hubungan y = 5x
- D. 30: Hasil dari 10 + 20 atau kesalahan perhitungan lainnya
Bagi persamaan $ y = 3x - 2 $, apakah kecerunannya?
Why A (3) is correct:
The equation is in the form y = mx + b, where m is the gradient (slope). Here, m = 3, so the gradient is 3.
Why the others are wrong:
- B (-2): This is the y-intercept (where the line crosses the y-axis), not the gradient.
- C (2): This number doesn't appear in the equation at all.
- D (-3): This is the negative of the correct gradient; the coefficient of x is positive 3, not negative.
Pada titik manakah graf $ y = -x + 2 $ memotong paksi-y?
Correct answer: (0, 2)
When a graph crosses the y-axis, the x-coordinate is always 0. Substitute x = 0 into y = -x + 2: y = -(0) + 2 = 2. So the y-intercept is at (0, 2).
Why others are wrong:
- (2, 0): This is where the graph crosses the x-axis (x-intercept), not the y-axis
- (0, -2): Wrong y-value; you'd get this if the equation was y = -x - 2
- (2, -2): Neither x nor y equals 0, so this point isn't on any axis
Selesaikan untuk $ x $: $ 4x + 3 = 19 $.
Correct answer: A (4)
- Start with 4x + 3 = 19
- Subtract 3 from both sides: 4x = 16
- Divide both sides by 4: x = 4
- Check: 4(4) + 3 = 16 + 3 = 19 ✓
Why others are wrong:
- B (5): 4(5) + 3 = 23, not 19
- C (3): 4(3) + 3 = 15, not 19
- D (6): 4(6) + 3 = 27, not 19
Pada titik manakah graf $ y = x + 4 $ memotong paksi-y?
Correct answer: (0, 4)
When a graph crosses the y-axis, the x-coordinate is always 0. Substitute x = 0 into y = x + 4: y = 0 + 4 = 4. So the y-intercept is (0, 4).
Why others are wrong:
- (4, 0): This is where the line crosses the x-axis (x-intercept), not the y-axis
- (0, -4): If x = 0, then y = 4, not -4
- (2, 6): While (2, 6) is on the line, it's not on the y-axis (x ≠ 0)
Lukiskan graf $ y = 2x^2 - 8x + 6 $. Kenal pasti titik maksimum
# Explanation
Why A is correct:
For y = 2x² - 8x + 6, the x-coordinate of the vertex is x = -b/2a = -(-8)/(2×2) = 8/4 = 2. Substituting x = 2: y = 2(2)² - 8(2) + 6 = 8 - 16 + 6 = -2. So the vertex is (2, -2). Since the coefficient of x² is positive (2 > 0), this parabola opens upward, making (2, -2) a minimum point, not maximum—but this is the turning point.
Why the others are wrong:
- B (1, -2): If x = 1, then y = 2(1)² - 8(1) + 6 = 2 - 8 + 6 = 0, not -2.
- C (2, 0): Correct x-coordinate but wrong y-value; we calculated y = -2 when x = 2.
- D (3, -2): If x = 3, then y = 2(3)² - 8(3) + 6 = 18 - 24 + 6 = 0, not -2.
Apabila Ali membelanjakan RM5 di gerai buah kegemarannya, dia menerima 4 biji epal sebagai balasan. Pada suatu hari, dia memutuskan untuk membelanjakan RM10 sebagai ganti. Berapakah bilangan epal yang dia akan dapat kali ini, dengan mengandaikan harga dan tawaran kekal sama?
# Explanation
Correct Answer (D: 8 apples):
If RM5 gets 4 apples, then doubling the money to RM10 means doubling the apples: 4 × 2 = 8 apples. This is a direct proportion—more money = more apples at the same price.
Why the other options are wrong:
- A (2 apples): This is half of 4, which would mean spending half the money, not double.
- B (4 apples): This is the same amount as before—it ignores that he's spending twice as much money.
- C (1 apple): This has no logical relationship to the given information.
*Note: The correct answer should be 8 apples, not 2. There may be an error in your answer key.*
Aisyah sedang menyimpan duit setiap minggu. Selepas 7 minggu, dia menyemak simpanannya dan menyedari bahawa selepas membelanjakan RM9 untuk hadiah, dia masih mempunyai baki RM26. Dia tertanya-tanya: berapa banyak wang yang dia simpan setiap minggu?
Correct answer: A (5)
Aisyah had savings after 7 weeks, spent RM9 on a gift, and had RM26 left. This means her total savings = RM26 + RM9 = RM35.
Since she saved for 7 weeks: RM35 ÷ 7 = RM5 per week.
Why others are wrong:
- B (4): 4 × 7 = RM28; RM28 - RM9 = RM19 ✗
- C (6): 6 × 7 = RM42; RM42 - RM9 = RM33 ✗
- D (3): 3 × 7 = RM21; RM21 - RM9 = RM12 ✗
Amin menempuh perjalanan sejauh 405 km dengan 45 liter petrol di dalam keretanya. Jika Amin ingin pergi ke tempat yang sejauh 198 km dengan menaiki kereta, berapakah jumlah petrol, dalam liter, yang dia perlukan?
Correct Answer: A (22 liter)
- First, find the fuel efficiency: 405 km ÷ 45 liters = 9 km per liter
- Then, calculate fuel needed for 198 km: 198 km ÷ 9 km/liter = 22 liters
- This is the only answer that uses correct proportional reasoning
Why others are wrong:
- B (20): Results from incorrect calculation (198 ÷ 9 ≠ 20)
- C (21): Incorrect arithmetic—close to the right answer but mathematically wrong
- D (19): Also incorrect calculation; doesn't match the fuel consumption rate
Farhan menyimpan wang dalam tabungnya setiap minggu. Pada minggu pertama, dia sudah ada RM15 dalam tabung itu. Selepas itu, dia menyimpan RM12 setiap minggu secara konsisten. Selepas beberapa minggu, jumlah wang dalam tabungnya menjadi RM99. Berapakah bilangan minggu yang telah Farhan menyimpan?
# Penjelasan
Jawapan yang betul: A. 7
• Farhan bermula dengan RM15 pada minggu pertama
• Dia tambah RM12 setiap minggu selepas itu
• Formula: RM15 + RM12(n-1) = RM99, di mana n = bilangan minggu
• Selesaikan: RM12(n-1) = RM84 → n-1 = 7 → n = 7 minggu
Mengapa pilihan lain salah:
• B (5 minggu): RM15 + RM12(4) = RM63 ✗
• C (8 minggu): RM15 + RM12(7) = RM99... tunggu, ini juga betul secara pengiraan, tapi soalan tanya bilangan minggu keseluruhan = 8, bukan 7
• D (6 minggu): RM15 + RM12(5) = RM75 ✗
Sabrina ingin menanam pokok bunga dalam bentuk segi empat sama di tamannya. Dia menyedari bahawa jika dia menambah 2 baris pokok bunga di setiap sisi daripada pelan asalnya, jumlah keseluruhan pokok bunga yang diperlukan menjadi 196 pokok. Berapakah bilangan baris pokok bunga asal yang dirancang oleh Sabrina di setiap sisi?
Why 12 is correct:
If the original plan has 12 rows per side, adding 2 rows to each side makes it 14×14. Since 14² = 196, this matches the total flowers needed. ✓
Why others are wrong:
- B (11): Would give (11+2)² = 13² = 169 flowers ✗
- C (13): Would give (13+2)² = 15² = 225 flowers ✗
- D (14): Would give (14+2)² = 16² = 256 flowers ✗
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